$ \left(\dfrac{1}{4}\right)^{-\frac{7}{2}}$
Explanation: $= 4^{\frac{7}{2}}$ $= \left(4^{\frac{1}{2}}\right)^{7}$ To simplify $4^{\frac{1}{2}}$ , figure out what goes in the blank: $\left(? \right)^{2}=4$ To simplify $4^{\frac{1}{2}}$ , figure out what goes in the blank: $\left({2}\right)^{2}=4$ so $ 4^{\frac{1}{2}}=2$ So $4^{\frac{7}{2}}=\left(4^{\frac{1}{2}}\right)^{7}=2^{7}$ $= 2\cdot2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2$ $= 4\cdot2\cdot 2\cdot 2\cdot 2\cdot 2$ $= 8\cdot2\cdot 2\cdot 2\cdot 2$ $= 16\cdot2\cdot 2\cdot 2$ $= 32\cdot2\cdot 2$ $= 64\cdot2$ $= 128$